Nonlinear Partial Differential Equations of Second Order (Translations of Mathematical Monographs)

This book addresses a class of equations central to many areas of mathematics and its applications. Although there is no routine way of solving nonlinear partial differential equations, effective approaches that apply to a wide variety of problems are available. This book addresses a general approach that consists of the following: Choose an appropriate function space, define a family of mappings, prove this family has a fixed point, and study various properties of the solution. The author emphasizes the derivation of various estimates, including a priori estimates. By focusing on a particular approach that has proven useful in solving a broad range of equations, this book makes a useful contribution to the literature. Cover S Title Nonlinear Partial Differential Equations of Second Order Copyright ©1991 by the American Mathematical Society ISBN 0-8218-4554-3 QA377.T8613 1991 515'.353-dc20 LCCN 91-27853 Contents Introduction Notes for the English Translation (1) The uniqueness and existence of solutions to nonlinear oblique derivative boundary value problems for fully nonlinear parabolic equations satisfying natural structure conditions (2) The existence, uniqueness, and regularity for viscosity solutions of thefirst boundary value problem for fully nonlinear parabolic equations satisfying anatural structure conditio (3) Fully nonlinear degenerate parabolic equations CHAPTER I The First Boundary Value Problemfor Second-Order Quasilinear Parabolic Equations with Principal Part in Divergence Form §1. Uniform and Holder estimates for the solut §2. A uniform bound for D_x u §3. A Holder estimate for D_x u §4. Existence and uniqueness of the solution for the first boundary value problem CHAPTER II A Periodic Boundary Value Problem for a Nonlinear Telegraph Equation §1. Solvability for higher-dimensional telegraph equations in the nonresonance case §2. A discussion on the resonance case §3. Regularity of a generalized solution CHAPTER III The Initial Value Problemfor a Nonlinear Schrodinger Equation §1. Background materials §2. The initial value problem for the linear Schrodinger equation §3. The initial value problem for a nonlinear Schrodinger equation CHAPTER IV Multi-Dimensional Subsonic Flows Around an Obstacle §1. Introduction §2. Background material for the linear problem §3. Solution to the auxiliary problem §4. Resolution of the problem of a flow passing an obstacle and elementary properties of the solution §5. Further properties of the solution CHAPTER V The Initial-Boundary Value Problem for Degenerate Quasilinear Parabolic Equations §1. Formulation of the problem and a Holder estimate for the solution §2. Solvability for the first boundary value problem §3. Uniqueness of the solution CHAPTER VI The Speed of Propagation of the Solution of a Degenerate Quasilinear Parabolic Equation §1. An estimate on the domain of dependence §2. A lower estimate for the solution CHAPTER VII Aleksandrov and Bony Maximum Principles for Parabolic Equations §1. Introduction §2. Some properties of convex functions §3. Convex envelopes §4. Several Aleksandrov maximum principles §5. Bony maximum principles CHAPTER VIII The Density Theorem and Its Applications §1. The statement of the density theorem §2. Several lemmas and the proof of the density theorem §3. The Harnack inequality for parabolic equations with measurable coefficients §4. A Holder estimate for the solution of a quasilinear parabolic equation §5. A Holder estimate for the solution for a quasilinear parabolic system CHAPTER IX Fully Nonlinear Parabolic Equations §1. A uniform bound for a solution u and an interior estimate for D_x u §2. An interior estimate for the second derivatives §3. An interior Holder estimate for the second derivatives §4. A near boundary Holder estimate for u §5. Uniform and Holder estimates for Dx u near the boundary §6. Near boundary uniform and Holder estimates for the second derivatives §7. Uniqueness and existence of a solution for the first boundary value problem under the natural structure conditions CHAPTER X Fully Nonlinear Parabolic Equations (Continued) §1. The density theorem for quasilinear parabolic equations with natural structure condition of the second kind §2. A Holder estimate for the solution and unique solvability for the first boundary value problem §3. Certain apriori estimates for the solutions of a fully nonlinear parabolic equation with natural structure condition of the second kind and the unique solvability of the first boundary value problem Symbols References Epilogue Back Cover The First Boundary Value Problem For Second Order Quasilinear Parabolic Equations With Principal Part In Divergence -- A Periodic Boundary Values Problem For A Nonlinear Telegraph Equation -- The Initial Value Problem For A Nonlinear Schrödinger Equation -- Multi-dimensional Subsonic Flows Around An Obstacle -- The Initial-boundary Value Problem For Degenerate Quasilinear -- The Speed Of Propagation Of The Solution Of A Degenerate Quasilinear Parabolic Equation -- Aleksandrovsk And Bony Maximum Principles For Parabolic Equations -- The Density Theorem And Its Applications -- Fully Nonlinear Parabolic Equations -- Fully Nonlinear Parabolic Equations. Guangchang Dong ; [translated From The Chinese By Kai Seng Chou (kaising Tso)]. Translation Of: Fei Hsien Hsing Erh Chieh Pʻien Wei Fen Fang Chʻeng. Original Chinese Title On Verso Of T.p.: Fei Hsien Hsing Erh Chieh Pʻien Wei Fen Fang Chʻeng. Includes Bibliographical References.